Linear Inequalities

 Inequalities  Operations on integers, Order of real number  In this lesson, you are expected to: 1. Differentiate between equations and inequalities, 2. Find the solution of an inequality involving one variable.

Differences between Linear Equation and Linear Inequalities:

Symbols Number of Solution Graph of Solution Example:

= Exactly one / one / infinite A point x + 7 = 25

≠ , , ≤, ≥ More than one Set of points 3x + 4 > 16

Inequality – A mathematical statement where one algebraic expression is not equal to another algebraic expression. ≠ < > ≤ ≥

Not equal Less than Greater than Less than or equal to Greater than or equal to

x+5≤6 7 (x + 2) < - 5 5 > 2x + 4 4 ≠ 5x + 2 x+2≥7

x > -4

-10

-5

0

5

10

-10

-5

0

5

10

-10

-5

0

5

10

-10

-5

0

5

10

-10

-5

0

5

10

-10

-5

0

5

10

x≠8

x≥9

x≤

2 3

-3 < x < 3

0≤x 7 2. x < 14 3. -4 ≥ 5

4. 10 ≤ 25 5. x ≠ -12

let a, b, and c are real numbers. If 2. x - 8 < 11 Solution: x – 8 < - 11 x – 8  < - 11 x < -3

, then

, and

1. x + 7 ≥ 0 Solution:

.

x + 7 ≥ 0 ≥ 0 x+7 x ≥ - 7

let a, b, and c are real numbers and c ≠ 0.

and and

 

and

and

 

and

and

 

and

3. Given: 5x – 9 > -2x + 5 Solution: 5x – 9  > -2x + 5 5x > -2x + 14 5x  > -2x  + 14 









and

4.

      



>



>

  

<



>

 

Given: 2(x - 3) ≤ 5x +12 ≤ 5x + 12 Solution: 2x - 6 2x ≤ 5x + 18 ≤ 5x 2x + 18 







(- ) -3x ≤ 18 (- )

( )7x > 14( )

x≥-6

x>2

≥ to ≤ and vice versa,

> to < and vice versa.

x>2: -10

-5

0

5

10

-10

-5

0

5

10

x ≥ - 6:

Given: Solution:

-3< x + 2 < 14 -3 < x + 2  < 14 -5 < x < 12

Given: Solution:

-1 ≤ 2 – 3x < 11 -1 ≤ 2  – 3x < 11 -3

 

≤ – 3x

 



-5 < x < 12:

1 ≥  x > -3: -3 < x ≤ 1

-10

-5

0

10

15

-10

-5

0

5

10

1. 2. 3. 4. 5.

x+4 x-5 x+5 x – 6 x+8

7 14 12 11 -4

= = = = =

6. 7. 8. 9. 10.

x – 9 2x 3x -5x -4x

-13 18 -9 25 24

= = = = =

1. 2. 3. 4. 5.

x + 47 y - 83 -2 + 9x 14 - 18y 15x - 4

6. 7. 8. 9. 10.

13 - y 29 + 2y 5x + 7 3(x + 1) -23 3x – 11 -2 -7 x + 14 8 -1 + 2y 5 - 2y

36 -79 10x -7y   11x – 16

= = = = = = = = = =